muHVT: Predicting Cells with Layers using predictLayerHVT

Zubin Dowlaty, Srinivasan Sudarsanam, Somya Shambhawi

2023-06-27

1 Abstract

The muHVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data analysis. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

  1. Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.

  2. Data Projection: Dimension projection of the compressed cells to 1D,2D or 3D with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension.

  3. Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.

  4. Prediction: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

2 Example: muHVT with the Personal Computer dataset

Data Understanding

In this vignette, we will use the Prices of Personal Computers dataset. This dataset contains 6259 observations and 6 features. The dataset observes the price from 1993 to 1995 of 486 personal computers in the US. The variables are price, speed, hd, ram, screen and ads.

Here, we load the data and store into a variable computers.

set.seed(240)
# Load data from csv files
computers <- read.csv("https://raw.githubusercontent.com/Mu-Sigma/muHVT/master/vignettes/sample_dataset/Computers.csv")

Raw Personal Computers Dataset

The Computers dataset includes the following columns:

Let’s explore the Personal Computers Dataset containing (6259 points). For the shake of brevity we are displaying first six rows.

Table(head(computers), scroll = T, limit = 20)
X price speed hd ram screen cd multi premium ads trend
1 1499 25 80 4 14 no no yes 94 1
2 1795 33 85 2 14 no no yes 94 1
3 1595 25 170 4 15 no no yes 94 1
4 1849 25 170 8 14 no no no 94 1
5 3295 33 340 16 14 no no yes 94 1
6 3695 66 340 16 14 no no yes 94 1

Now, let us check the structure of the data and analyse its summary.

str(computers)
#> 'data.frame':    6259 obs. of  11 variables:
#>  $ X      : int  1 2 3 4 5 6 7 8 9 10 ...
#>  $ price  : int  1499 1795 1595 1849 3295 3695 1720 1995 2225 2575 ...
#>  $ speed  : int  25 33 25 25 33 66 25 50 50 50 ...
#>  $ hd     : int  80 85 170 170 340 340 170 85 210 210 ...
#>  $ ram    : int  4 2 4 8 16 16 4 2 8 4 ...
#>  $ screen : int  14 14 15 14 14 14 14 14 14 15 ...
#>  $ cd     : chr  "no" "no" "no" "no" ...
#>  $ multi  : chr  "no" "no" "no" "no" ...
#>  $ premium: chr  "yes" "yes" "yes" "no" ...
#>  $ ads    : int  94 94 94 94 94 94 94 94 94 94 ...
#>  $ trend  : int  1 1 1 1 1 1 1 1 1 1 ...
summary(computers)
#>        X            price          speed              hd        
#>  Min.   :   1   Min.   : 949   Min.   : 25.00   Min.   :  80.0  
#>  1st Qu.:1566   1st Qu.:1794   1st Qu.: 33.00   1st Qu.: 214.0  
#>  Median :3130   Median :2144   Median : 50.00   Median : 340.0  
#>  Mean   :3130   Mean   :2220   Mean   : 52.01   Mean   : 416.6  
#>  3rd Qu.:4694   3rd Qu.:2595   3rd Qu.: 66.00   3rd Qu.: 528.0  
#>  Max.   :6259   Max.   :5399   Max.   :100.00   Max.   :2100.0  
#>       ram             screen           cd               multi          
#>  Min.   : 2.000   Min.   :14.00   Length:6259        Length:6259       
#>  1st Qu.: 4.000   1st Qu.:14.00   Class :character   Class :character  
#>  Median : 8.000   Median :14.00   Mode  :character   Mode  :character  
#>  Mean   : 8.287   Mean   :14.61                                        
#>  3rd Qu.: 8.000   3rd Qu.:15.00                                        
#>  Max.   :32.000   Max.   :17.00                                        
#>    premium               ads            trend      
#>  Length:6259        Min.   : 39.0   Min.   : 1.00  
#>  Class :character   1st Qu.:162.5   1st Qu.:10.00  
#>  Mode  :character   Median :246.0   Median :16.00  
#>                     Mean   :221.3   Mean   :15.93  
#>                     3rd Qu.:275.0   3rd Qu.:21.50  
#>                     Max.   :339.0   Max.   :35.00

Let us first split the data into train and test. We will randomly select 80% of the data for training and remaining as testing.


num_rows <- nrow(computers)
set.seed(123)
train_indices <- sample(1:num_rows, 0.8 * num_rows)
trainComputers <- computers[train_indices, ]
testComputers <- computers[-train_indices, ]

K-means is not suitable for factor variables as the sample space for factor variables is discrete. A Euclidean distance function on such a space isn’t really meaningful. Hence, we will delete the factor variables(X, cd, multi, premium, trend) in our dataset.

trainComputers <-
  trainComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))
testComputers <-
  testComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))

Raw Training Dataset

Now, lets have a look at the randomly selected raw training dataset containing (5007 data points). For the sake of brevity we are displaying first six rows.

trainComputers_data <- trainComputers %>% as.data.frame() %>% round(4)
trainComputers_data$Row.No <- as.numeric(row.names(trainComputers_data))
trainComputers_data <- trainComputers_data %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
row.names(trainComputers_data) <- NULL
Table(head(trainComputers_data))
Row.No price speed hd ram screen ads
2463 2799 50 230 8 15 216
2511 2197 33 270 4 14 216
2227 2744 50 340 8 17 275
526 2999 66 245 16 15 139
4291 1974 33 200 4 14 248
2986 2490 33 528 16 14 267

Raw Testing Dataset

Now, lets have a look at the randomly selected raw testing dataset containing (1252 data points). For the sake of brevity we are displaying first six rows.

#testComputers <- scale(testComputers, center = scale_attr$`scaled:center`, scale = scale_attr$`scaled:scale`) 
testComputers_data <- testComputers %>% as.data.frame() %>% round(4)
testComputers_data$Row.No <- as.numeric(row.names(testComputers_data))
testComputers_data <- testComputers_data %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
rownames(testComputers_data) <- NULL
Table(head(testComputers_data))
Row.No price speed hd ram screen ads
3 1595 25 170 4 15 94
4 1849 25 170 8 14 94
7 1720 25 170 4 14 94
10 2575 50 210 4 15 94
11 2195 33 170 8 15 94
14 2295 25 245 8 14 94

3 Map A : Base Compressed Map

Let us try to visualize the compressed Map A from the flow diagram below.

Figure 1: Flow map with highlighted bounding box in red around compressed map A

Figure 1: Flow map with highlighted bounding box in red around compressed map A

This package can perform vector quantization using the following algorithms -

For more information on vector quantization, refer the following link.

The HVT function constructs highly compressed hierarchical Voronoi tessellations. The raw data is first scaled and this scaled data is supplied as input to the vector quantization algorithm. The vector quantization algorithm compresses the dataset until a user-defined compression percentage/rate is achieved using a parameter called quantization error which acts as a threshold and determines the compression percentage. It means that for a given user-defined compression percentage we get the ‘n’ number of cells, then all of these cells formed will have a quantization error less than the threshold quantization error.

Let’s try to comprehend the HVT function first before moving ahead.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  normalize = TRUE,
  diagnose = FALSE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters have been explained below :

We will use the HVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the HVT function again.

In our example we will iteratively increase the number of cells until the desired compression percentage is reached instead of increasing the quantization threshold because it may reduce the level of detail captured in the data representation

First, we will construct map A by using the below mentioned model parameters.

3.0.1 Iteration 1:

We will pass the below mentioned model parameters along with training dataset to HVT function.

Model Parameters

set.seed(240)
map_A <- list()
map_A  <- muHVT::HVT(
  trainComputers,
  n_cells = 200,
  depth = 1,
  quant.err = 0.2,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary.

compressionSummaryTable(map_A[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 200 83 0.42 n_cells: 200 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 42% of cells have reached the quantization threshold error. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if desired compression (80%) is reached

3.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

map_A <- list()
map_A <-muHVT::HVT(trainComputers,
                n_cells = 440,
                quant.err = 0.2,
                depth = 1,
                distance_metric = "L1_Norm",
                error_metric = "max",
                quant_method = "kmeans",
                normalize = T)

As per the manual, map_A[[3]] gives us detailed information about the hierarchical vector quantized data. map_A[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from map A (containing 440 records)

summaryTable(map_A[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 7 46 0.08 -0.76 -0.89 -0.88 -0.76 -0.67 1.57
1 1 2 10 108 0.08 -0.80 -0.89 -0.16 -0.76 -0.67 0.67
1 1 3 15 223 0.12 0.37 -0.89 -0.72 -0.05 0.43 -1.65
1 1 4 11 54 0.07 -1.50 -0.89 -0.75 -0.76 -0.67 0.62
1 1 5 8 146 0.13 -0.31 0.68 -0.95 -0.89 -0.67 -0.14
1 1 6 11 150 0.16 -0.66 0.68 -0.78 -0.79 -0.67 -0.73
1 1 7 11 170 0.1 0.03 -1.24 -0.13 -0.05 -0.67 0.38
1 1 8 8 334 0.15 0.62 2.30 0.08 -0.05 0.43 0.04
1 1 9 8 114 0.07 -0.16 0.68 -1.19 -1.11 -0.67 0.87
1 1 10 7 248 0.17 0.51 -0.08 0.34 -0.05 -0.67 -0.30
1 1 11 9 140 0.12 -0.01 0.68 -1.15 -1.00 -0.67 0.33
1 1 12 7 219 0.14 -1.36 0.24 0.46 -0.05 -0.67 -0.74
1 1 13 9 271 0.05 -1.08 0.68 0.49 -0.05 0.43 -0.84
1 1 14 19 109 0.06 -0.31 -0.89 -0.74 -0.76 -0.67 0.38
1 1 15 6 176 0.08 -0.72 -0.89 -0.07 -0.05 0.43 0.72
1 1 16 17 332 0.14 0.42 2.30 0.10 -0.05 0.43 1.50
1 1 17 12 18 0.05 -1.21 -1.27 -1.19 -1.11 -0.67 0.97
1 1 18 19 149 0.16 -0.68 -0.08 -0.46 -0.76 0.43 0.79
1 1 19 17 428 0.35 0.18 2.30 2.53 1.37 0.43 -2.22
1 1 20 20 320 0.36 0.82 -0.16 -0.09 -0.12 2.64 0.71
1 1 21 3 305 0.18 2.27 -0.35 -0.01 -0.05 -0.67 -1.32
1 1 22 7 227 0.1 -0.51 -0.89 0.45 -0.05 0.43 -0.47
1 1 23 10 178 0.12 0.00 0.68 -0.86 -0.76 -0.67 -0.90
1 1 24 9 365 0.1 0.68 -0.08 1.20 1.37 0.43 -0.36
1 1 25 5 14 0.11 -1.99 -0.89 -0.96 -1.11 -0.67 0.18
1 1 26 3 411 0.05 1.25 -0.89 2.29 2.79 0.43 0.57
1 1 27 18 122 0.15 -0.18 -0.98 -0.85 -0.76 0.43 0.68
1 1 28 15 189 0.11 0.40 -0.92 0.03 -0.05 -0.67 0.87
1 1 29 11 107 0.11 -0.49 -0.96 -0.88 -0.76 -0.67 -0.64
1 1 30 7 423 0.47 3.55 0.12 2.51 1.37 -0.67 0.44
1 1 31 14 90 0.05 -0.63 -0.89 -0.79 -0.76 -0.67 0.58
1 1 32 22 430 0.24 0.63 0.75 3.07 2.79 0.43 -2.27
1 1 33 5 390 0.3 1.37 -0.89 3.73 -0.19 -0.45 0.70
1 1 34 25 101 0.18 -0.85 -0.97 -0.71 -0.76 0.43 0.84
1 1 35 11 425 0.07 0.15 2.30 1.70 1.37 0.43 -2.39
1 1 36 10 358 0.05 0.24 -0.89 1.20 1.37 0.43 -0.84
1 1 37 16 166 0.11 0.03 0.68 -1.08 -0.78 -0.67 -1.65
1 1 38 13 45 0.05 -0.91 -0.89 -1.19 -1.11 -0.67 0.42
1 1 39 8 383 0.12 1.15 2.30 0.45 1.37 -0.67 -0.16
1 1 40 5 9 0.07 -1.24 -0.97 -1.19 -1.11 -0.67 1.57
1 1 41 11 419 0.06 1.41 -0.89 2.29 2.79 0.43 -0.82
1 1 42 8 242 0.14 -0.81 -0.08 0.30 -0.05 0.43 -0.65
1 1 43 13 179 0.09 0.41 0.68 -0.76 -0.76 -0.67 0.40
1 1 44 5 375 0.04 0.06 -0.08 1.70 1.37 0.43 -0.79
1 1 45 20 129 0.14 -1.15 0.68 -0.79 -0.76 -0.67 -0.40
1 1 46 10 292 0.22 0.86 0.68 -0.65 -0.12 0.43 -1.41
1 1 47 5 79 0.12 -0.89 -1.04 -0.94 -0.05 -0.67 1.02
1 1 48 23 246 0.11 -0.42 0.68 0.46 -0.05 -0.67 -0.63
1 1 49 8 207 0.25 0.74 -0.89 -0.40 -0.40 0.43 0.52
1 1 50 11 27 0.06 -1.06 -1.27 -1.19 -1.11 -0.67 0.43
1 1 51 8 51 0.11 -1.23 -0.08 -1.05 -0.85 -0.67 0.94
1 1 52 19 288 0.09 0.81 -0.89 0.45 1.37 -0.67 0.88
1 1 53 7 154 0.1 -0.62 0.68 -0.15 -0.76 -0.67 0.84
1 1 54 10 261 0.15 0.61 -0.08 -0.67 -0.05 0.43 -1.40
1 1 55 10 195 0.15 0.18 0.77 -0.09 -0.76 -0.67 0.83
1 1 56 14 250 0.09 0.52 0.68 -0.69 -0.05 -0.67 -1.61
1 1 57 20 331 0.2 -0.63 0.68 0.30 -0.76 2.64 -0.95
1 1 58 9 379 0.15 1.33 -0.08 -0.65 -0.29 2.64 -1.52
1 1 59 14 11 0.21 -0.28 -1.05 -0.79 -0.76 2.64 0.53
1 1 60 29 359 0.13 1.36 0.68 0.18 1.37 0.43 0.77
1 1 61 6 337 0.1 2.46 0.68 0.21 -0.05 -0.67 -0.87
1 1 62 6 1 0.17 -0.17 -1.21 -1.02 -0.76 2.64 1.32
1 1 63 28 243 0.33 -0.33 2.30 -0.23 -0.46 -0.67 -0.88
1 1 64 8 274 0.07 0.41 -1.27 0.45 1.37 -0.67 0.68
1 1 65 13 362 0.14 1.07 0.75 0.35 1.37 0.43 1.34
1 1 66 10 143 0.07 -0.34 -0.89 -0.80 -0.05 -0.67 -1.66
1 1 67 4 265 0.05 -0.55 0.68 1.23 -0.05 -0.67 -0.69
1 1 68 11 13 0.15 -0.83 -0.89 -0.25 -0.76 2.64 -0.33
1 1 69 8 298 0.17 -0.62 0.20 2.29 -0.05 -0.67 -0.95
1 1 70 4 335 0.06 1.34 -0.08 0.45 1.37 -0.67 -0.08
1 1 71 20 204 0.16 0.09 -0.08 0.02 -0.05 -0.67 0.86
1 1 72 10 42 0.06 -1.49 -0.89 -0.75 -0.76 -0.67 1.04
1 1 73 1 429 0 3.08 0.68 0.04 4.20 0.43 0.71
1 1 74 14 186 0.14 -0.79 -0.89 0.45 -0.05 -0.67 -0.68
1 1 75 4 410 0.37 2.27 0.68 3.73 -0.23 -0.40 0.68
1 1 76 9 163 0.16 1.05 -0.89 -0.41 -0.60 -0.67 0.61
1 1 77 10 400 0.07 -0.03 0.68 1.70 1.37 0.43 -2.38
1 1 78 6 275 0.18 1.14 0.68 0.13 -0.05 -0.67 -0.18
1 1 79 25 241 0.16 -0.88 0.68 0.40 -0.05 -0.67 -1.06
1 1 80 6 245 0.14 -1.22 0.68 -0.30 -0.05 0.43 -0.91
1 1 81 21 120 0.16 -0.46 -0.08 -0.82 -0.62 -0.67 0.70
1 1 82 11 40 0.18 -0.93 -0.99 -1.19 -1.11 0.43 0.37
1 1 83 9 342 0.28 1.16 0.43 -0.52 -0.68 2.64 1.12
1 1 84 8 286 0.05 -0.72 1.11 0.50 -0.05 0.43 -0.83
1 1 85 8 33 0.1 -1.06 -0.89 -1.18 -1.02 -0.67 -0.99
1 1 86 5 282 0.23 -1.50 0.53 0.19 -0.48 0.43 -2.16
1 1 87 17 137 0.07 -0.31 0.68 -0.78 -0.76 -0.67 0.96
1 1 88 19 168 0.07 -0.08 -0.89 0.05 -0.05 -0.67 1.04
1 1 89 7 291 0.05 1.10 -0.89 0.15 1.37 -0.67 0.36
1 1 90 6 24 0.19 -0.97 -1.02 -1.07 -0.94 0.43 -1.40
1 1 91 4 434 0.59 4.07 1.49 1.29 0.30 2.64 0.07
1 1 92 19 409 0.65 0.92 0.42 1.41 1.37 2.64 -0.60
1 1 93 9 393 0.46 2.04 2.30 0.91 -0.05 -0.31 -0.34
1 1 94 8 22 0.08 -1.66 -1.27 -0.84 -0.76 -0.67 0.84
1 1 95 23 158 0.22 -1.41 0.68 -0.20 -0.70 -0.67 -1.05
1 1 96 11 330 0.22 1.55 0.68 -0.45 -0.31 0.43 -1.59
1 1 97 6 145 0.17 0.49 -0.89 -0.64 -0.76 -0.67 -0.17
1 1 98 12 121 0.15 0.21 -0.89 -0.80 -0.76 -0.67 -1.70
1 1 99 14 329 0.24 2.06 0.63 0.31 -0.25 0.43 0.99
1 1 100 16 299 0.05 1.21 -0.89 0.46 1.37 -0.67 0.86
1 1 101 5 328 0.1 -0.87 1.11 1.33 -0.05 0.43 -1.13
1 1 102 10 278 0.1 0.33 -0.93 0.45 1.37 -0.67 0.02
1 1 103 5 102 0.1 -1.05 0.68 -0.76 -0.76 -0.67 1.27
1 1 104 12 25 0.08 -1.25 -1.05 -0.78 -0.76 -0.67 1.57
1 1 105 9 385 0.25 2.05 0.34 -0.21 -0.05 2.64 1.12
1 1 106 10 231 0.18 -0.02 -0.08 0.02 -0.05 0.43 1.29
1 1 107 5 193 0.14 -0.66 -0.08 -0.33 -0.05 -0.67 -1.01
1 1 108 4 418 0.06 -0.03 0.68 3.07 1.37 0.43 -2.25
1 1 109 4 306 0.16 1.83 0.68 -0.59 -0.05 -0.67 -1.57
1 1 110 5 378 0.15 1.09 0.68 1.20 1.37 0.43 -0.11
1 1 111 2 308 0.07 -0.55 0.68 0.26 1.37 -0.67 -1.25
1 1 112 18 239 0.15 0.44 0.75 0.08 -0.05 -0.67 1.05
1 1 113 11 62 0.06 -1.14 -0.89 -0.81 -0.76 -0.67 0.73
1 1 114 5 169 0.07 -0.26 -0.89 0.47 -0.05 -0.67 1.40
1 1 115 18 415 0.11 0.74 -0.08 2.29 2.79 0.43 -0.93
1 1 116 5 97 0.13 0.23 -1.19 -1.05 -0.76 -0.67 0.77
1 1 117 24 19 0.22 -0.06 2.30 -0.89 -0.89 -0.67 1.18
1 1 118 9 184 0.17 -0.98 0.34 -0.29 -0.05 -0.67 -0.08
1 1 119 11 43 0.12 -0.82 -0.89 -1.04 -0.79 -0.67 -1.65
1 1 120 9 257 0.29 2.17 -0.62 0.10 -0.05 -0.55 0.65
1 1 121 27 155 0.18 0.15 0.68 -0.84 -0.76 -0.67 0.81
1 1 122 20 215 0.11 0.33 -0.08 -0.69 -0.05 -0.67 -1.64
1 1 123 14 348 0.24 0.61 0.74 0.37 1.37 0.43 0.25
1 1 124 13 162 0.12 0.01 -1.01 -0.67 -0.05 -0.67 -1.57
1 1 125 7 366 0.22 1.62 0.35 -0.29 1.37 -0.67 -1.65
1 1 126 9 77 0.02 -0.88 -0.89 -0.78 -0.76 -0.67 0.66
1 1 127 25 253 0.14 0.70 0.68 0.16 -0.05 -0.67 0.61
1 1 128 9 84 0.09 -0.27 -1.27 -0.81 -0.76 -0.67 0.81
1 1 129 3 398 0.04 -0.03 -0.08 2.29 1.37 0.43 -1.98
1 1 130 9 412 0.05 0.90 -0.89 2.29 2.79 0.43 -0.48
1 1 131 8 309 0.06 1.43 -0.89 0.41 1.37 -0.67 0.46
1 1 132 8 267 0.08 0.10 0.68 0.37 -0.05 0.43 1.57
1 1 133 16 252 0.22 0.94 -0.08 -0.40 -0.14 0.43 0.43
1 1 134 15 119 0.13 -0.60 -0.99 -0.75 -0.76 0.43 0.24
1 1 135 13 48 0.09 -0.66 -0.92 -1.19 -1.11 -0.67 0.82
1 1 136 9 326 0.11 1.45 -0.08 0.32 1.37 -0.67 0.38
1 1 137 11 386 0.22 1.37 0.68 -0.66 -0.18 2.64 -1.36
1 1 138 20 255 0.15 0.43 -0.89 -0.34 -0.05 2.64 0.68
1 1 139 8 161 0.14 -0.68 -0.94 -0.08 -0.05 0.43 1.41
1 1 140 10 38 0.13 -1.22 -1.19 -0.92 -0.76 0.43 0.91
1 1 141 14 180 0.12 0.09 0.68 -0.56 -0.76 -0.67 0.07
1 1 142 13 427 0.21 2.01 0.62 2.29 2.79 0.43 -0.18
1 1 143 15 354 0.13 1.63 0.68 0.39 1.37 -0.67 0.26
1 1 144 11 301 0.14 0.82 -1.03 0.15 1.37 -0.67 -0.85
1 1 145 7 205 0.13 0.24 -0.08 -0.05 -0.05 -0.67 1.57
1 1 146 10 327 0.14 0.66 0.68 0.60 1.37 -0.67 0.42
1 1 147 11 403 0.11 0.92 2.30 1.22 1.37 0.43 -0.79
1 1 148 7 127 0.08 -0.20 -0.89 -0.10 -0.76 -0.67 0.69
1 1 149 6 217 0.11 0.41 -0.89 0.26 -0.05 -0.67 -0.34
1 1 150 11 323 0.1 0.96 -0.89 0.46 1.37 0.43 0.72
1 1 151 9 208 0.16 0.16 0.68 -0.60 -0.05 -0.67 0.43
1 1 152 14 405 0.14 -0.04 0.68 2.29 1.37 0.43 -2.24
1 1 153 10 98 0.08 -0.22 -0.89 -1.14 -0.76 -0.67 0.33
1 1 154 7 47 0.05 -0.99 -0.89 -1.16 -0.76 -0.67 1.03
1 1 155 8 182 0.15 -1.00 0.68 0.08 -0.76 -0.67 -0.46
1 1 156 9 153 0.07 -0.24 -0.93 0.04 -0.05 -0.67 1.57
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1 1 415 8 437 0.27 1.02 1.00 3.07 2.79 2.64 -2.30
1 1 416 7 284 0.19 0.20 -0.66 -0.74 -0.05 2.64 -0.71
1 1 417 9 264 0.06 0.29 -0.89 0.45 1.37 -0.67 0.66
1 1 418 21 238 0.21 0.24 -0.08 -0.09 -0.05 0.43 0.69
1 1 419 3 296 0.02 -0.47 0.68 0.03 -0.05 0.43 -2.34
1 1 420 10 100 0.16 -0.44 -0.97 -0.87 -0.76 0.43 1.21
1 1 421 11 174 0.15 -0.61 0.68 -0.02 -0.76 -0.67 0.17
1 1 422 9 311 0.25 1.52 -0.53 -0.55 -0.76 2.64 0.36
1 1 423 6 135 0.09 -0.58 -1.27 0.12 -0.05 -0.67 1.29
1 1 424 5 368 0.15 2.37 0.68 0.36 1.37 -0.67 0.68
1 1 425 9 373 0.12 1.65 0.68 0.39 1.37 0.43 -0.01
1 1 426 10 228 0.13 0.42 -0.08 -0.33 -0.05 -0.67 -0.92
1 1 427 20 86 0.22 -1.23 -0.93 -0.85 -0.83 0.43 -0.46
1 1 428 7 106 0.06 -0.07 -0.89 -0.76 -0.76 -0.67 0.98
1 1 429 8 138 0.12 -0.45 -0.99 -0.78 -0.76 0.43 -0.77
1 1 430 8 30 0.12 -0.51 1.05 -1.19 -1.11 -0.67 1.36
1 1 431 19 357 0.16 1.03 0.68 -0.27 -0.05 2.64 0.61
1 1 432 20 406 0.38 0.02 2.30 0.99 -0.05 2.64 -1.12
1 1 433 10 302 0.26 1.83 0.53 -0.09 -0.33 0.43 0.24
1 1 434 5 73 0.04 -0.72 -0.89 -0.95 -0.76 -0.67 0.82
1 1 435 1 440 0 5.51 0.68 3.07 4.20 2.64 0.50
1 1 436 14 300 0.29 2.04 0.68 0.38 -0.15 -0.67 0.69
1 1 437 19 74 0.1 -1.10 -0.89 -0.81 -0.76 -0.67 -0.90
1 1 438 7 347 0.13 1.21 -0.08 0.41 1.37 0.43 0.59
1 1 439 31 374 0.28 -0.27 0.87 0.49 -0.05 2.64 -1.24
1 1 440 17 233 0.2 -0.48 0.68 -0.10 -0.13 0.43 0.60

Now let us understand what each column in the above table means:

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Now, let’s check the compression summary for HVT (map A). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapA_compression_summary <- map_A[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapA_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 440 355 0.81 n_cells: 440 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 81% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

Now let’s try to understand plotHVT function. The parameters have been explained in detail below:

plotHVT(hvt.results, line.width, color.vec, pch1 = 21, centroid.size = 3, title = NULL, maxDepth = 1)

Let’s plot the Voronoi tessellation for layer 1 (map A).

muHVT::plotHVT(map_A,
        line.width = c(0.4), 
        color.vec = c("#141B41"),
        centroid.size = 0.01,
        maxDepth = 1) 
Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 440 cells in the dataset ’computers’

Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 440 cells in the dataset ’computers’

Heat Maps

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data.

Let’s have a look at the function hvtHmap that we will use to overlay features as heatmap.

hvtHmap(hvt.results, dataset, child.level, hmap.cols, color.vec ,line.width, palette.color = 6)

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "n",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell

Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "price",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset

Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "speed",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset

Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "hd",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset

Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ram",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset

Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "screen",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset

Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ads",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

4 Map B : Compressed Novelty Map

Let us try to visualize the Map B from the flow diagram below.

Figure 10: Flow map with highlighted bounding box in red around map B

Figure 10: Flow map with highlighted bounding box in red around map B

In this section, we will manually figure out the novelty cells from the plotted map A and store it in identified_Novelty_cells variable.

Note: For manual selecting the novelty cells from map A, one can enhance its interactivity by adding plotly elements to the code. This will transform map A into an interactive plot, allowing users to actively engage with the data. By hovering over the centroids of the cells, a tag containing segment child information will be displayed. Users can explore the map by hovering over different cells and selectively choose the novelty cells they wish to consider. Added an image for reference.

Figure 11: Manually selecting novelty cells

Figure 11: Manually selecting novelty cells

The removeNovelty function removes the identified novelty cell(s) from the dataset and stores those records separately.

It takes input as the cell number (Segment.Child) of the manually identified novelty cell(s) from the above table and the compressed HVT map (map A). It returns a list of two items: dataset with novelty records, and a subset of the dataset without the novelty records.

identified_Novelty_cells <<- c(73,321,332,338,435)   #73,321,332,338,435
output_list <- removeNovelty(identified_Novelty_cells, map_A)

[1] “The following cell(s) have been removed as novelties from the dataset: 73 321 332 338 435”

data_with_novelty <- output_list[[1]]
dataset_without_novelty <- output_list[[2]]

Let’s have a look at the data with novelties(containing 24 records). For the sake of brevity, we will only show the first 10 rows.

novelty_data <- data_with_novelty
novelty_data$Row.No <- row.names(novelty_data)
novelty_data <- novelty_data %>% dplyr::select("Row.No","Cell.ID","Cell.Number","price","speed","hd","ram","screen","ads")
colnames(novelty_data) <- c("Row.No","Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
novelty_data %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = T, limit = 20)
Row.No Cell.ID Segment.Child price speed hd ram screen ads
1 429 73 3.0762240 0.6794579 0.0421969 4.2031619 0.4307274 0.7120258
2 438 321 2.6449847 0.6794579 6.5710368 1.3676416 0.4307274 0.6851382
3 438 321 1.6578103 2.2969425 6.5710368 1.3676416 0.4307274 -1.2507722
4 439 332 0.9130997 0.6794579 4.6232922 2.7854017 0.4307274 -2.4607161
5 439 332 1.8569770 1.1076156 4.6232922 4.2031619 2.6404120 -2.4607161
6 439 332 1.5192595 2.2969425 4.6232922 4.2031619 0.4307274 -2.4607161
7 439 332 1.4482522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
8 439 332 1.8569770 1.1076156 4.6232922 4.2031619 2.6404120 -2.4607161
9 439 332 1.1555636 2.2969425 4.6232922 2.7854017 0.4307274 -2.4607161
10 439 332 1.7773103 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
11 439 332 1.3460710 0.6794579 4.6232922 4.2031619 0.4307274 -2.4607161
12 439 332 1.4482522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
13 218 338 -1.4959522 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2993903
14 218 338 -1.5115391 0.6794579 0.4473277 -0.7589986 -0.6741149 -1.9767386
15 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2859465
16 218 338 -1.4959522 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.4472723
17 218 338 -1.5981334 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2859465
18 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2993903
19 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.4472723
20 218 338 -1.3227637 0.6794579 0.4473277 -0.7589986 -0.6741149 -1.9767386

4.1 Voronoi Tessellation with highlighted novelty cell

The plotCells function is used to plot the Voronoi tessellation using the compressed HVT map (map A) and highlights the identified novelty cell(s) in red on the map.

plotCells(identified_Novelty_cells, map_A,line.width = c(0.4),centroid.size = 0.01)
Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

We pass the dataframe with novelty records to HVT function along with other model parameters mentioned below to generate map B (layer2)

Model Parameters

colnames(data_with_novelty) <- c("Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
dataset_with_novelty <- data_with_novelty[,-1:-2]
map_B <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_B <- muHVT::HVT(dataset_with_novelty,
                  n_cells = 12 ,   #6
                  depth = 1,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F
                  )

The datatable displayed below is the summary from map B (layer 2) containing 12 records.

summaryTable(map_B[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 1 6 0 -1.08 0.68 0.45 -0.76 -0.67 -1.98
1 1 2 1 3 0 -1.51 0.68 0.45 -0.76 -0.67 -1.98
1 1 3 2 1 0.01 -1.55 -0.08 0.45 -0.76 -0.67 -1.98
1 1 4 2 11 0 1.86 1.11 4.62 4.20 2.64 -2.46
1 1 5 1 7 0 3.08 0.68 0.04 4.20 0.43 0.71
1 1 6 5 8 0.31 1.39 0.94 4.62 3.92 0.43 -2.46
1 1 7 1 12 0 5.51 0.68 3.07 4.20 2.64 0.50
1 1 8 2 9 0.15 1.34 2.30 4.62 3.49 0.43 -2.46
1 1 9 1 4 0 -1.32 0.68 0.45 -0.76 -0.67 -1.98
1 1 10 3 2 0.02 -1.53 0.68 0.45 -0.76 -0.67 -2.34
1 1 11 2 10 0.38 2.15 1.49 6.57 1.37 0.43 -0.28
1 1 12 3 5 0.02 -1.17 0.68 0.45 -0.76 -0.67 -2.34

Now let’s check the compression summary for HVT (map B). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapB_compression_summary <- map_B[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapB_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 12 10 0.83 n_cells: 12 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 83% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

5 Map C : Compressed Map without Novelty

Let us try to visualize the compressed Map C from the flow diagram below.

Figure 13:Flow map with highlighted bounding box in red around compressed map C

Figure 13:Flow map with highlighted bounding box in red around compressed map C

5.0.1 Iteration 1:

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the dataset without Novelty and below mentioned model parameters.

Model Parameters

map_C <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 10,
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 10 0 0 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 100 7 0.07 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 7% of the cells have hit the quantization threshold error in level 2

5.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression at depth 2. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

map_C <- list()
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 23,    #23
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

The datatable displayed below is the summary from map C (layer2) containing 552 records.

summaryTable(map_C[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 391 512 0.56 -1.27 -1.04 -0.91 -0.83 -0.59 0.86
1 1 2 105 96 1.02 2.08 0.75 0.11 0.71 2.64 0.56
1 1 3 190 55 1.37 1.48 0.02 2.32 2.63 0.12 -0.22
1 1 4 86 15 0.94 0.87 1.27 2.61 2.57 1.33 -1.76
1 1 5 257 326 0.49 0.41 0.45 -0.08 -0.20 -0.67 0.66
1 1 6 116 201 0.77 1.57 0.45 -0.24 0.27 -0.30 -1.30
1 1 7 235 228 0.58 0.85 -0.82 0.42 1.37 -0.47 0.52
1 1 8 149 185 0.84 -0.79 1.73 0.76 -0.22 -0.28 -1.75
1 1 9 353 306 0.69 -0.77 0.55 0.28 -0.22 -0.17 -0.79
1 1 10 234 297 0.54 0.25 0.50 -0.11 -0.16 0.43 0.66
1 1 11 330 478 0.56 -1.18 -0.79 -0.70 -0.76 -0.52 -0.69
1 1 12 373 434 0.6 -0.50 0.40 -0.84 -0.84 -0.56 0.66
1 1 13 209 147 0.81 1.32 0.99 0.36 1.37 -0.01 0.70
1 1 14 337 471 0.36 -0.51 -0.97 -0.87 -0.75 -0.67 0.57
1 1 15 283 404 0.71 0.06 -0.40 -0.78 -0.40 -0.42 -1.42
1 1 16 168 142 0.85 0.13 0.66 0.48 -0.02 2.64 -0.97
1 1 17 285 384 0.53 -0.04 -0.96 0.05 -0.12 -0.65 0.54
1 1 18 118 301 0.75 0.35 2.30 -0.13 -0.29 -0.28 0.82
1 1 19 256 414 0.47 -0.47 -0.84 -0.43 -0.48 0.43 0.60
1 1 20 78 57 1 0.18 1.67 2.05 1.31 0.39 -1.97
1 1 21 171 360 0.64 0.24 -0.46 -0.37 -0.41 2.64 0.59
1 1 22 170 130 1.28 0.28 -0.01 1.64 0.99 0.21 -0.62
1 1 23 89 219 0.88 1.98 0.59 0.20 -0.20 0.12 0.58
2 1 1 19 528 0.09 -1.31 -1.17 -1.19 -1.11 -0.67 0.99
2 1 2 18 479 0.31 -1.42 -0.87 -0.21 -0.72 -0.67 0.53
2 1 3 28 521 0.1 -1.47 -1.22 -1.11 -0.76 -0.67 0.89
2 1 4 10 530 0.14 -1.84 -1.04 -1.20 -1.11 -0.67 0.79
2 1 5 27 519 0.1 -1.20 -1.09 -1.19 -1.11 -0.67 0.50
2 1 6 13 524 0.18 -1.36 -1.10 -0.99 -0.87 0.43 1.36
2 1 7 5 523 0.04 -1.19 -1.27 -0.78 -0.76 -0.67 1.57
2 1 8 23 494 0.06 -1.18 -0.89 -0.78 -0.76 -0.67 0.72
2 1 9 27 514 0.14 -1.35 -1.27 -1.08 -0.76 -0.67 0.39
2 1 10 35 510 0.1 -1.08 -1.27 -0.88 -0.76 -0.67 0.90
2 1 11 6 529 0.15 -1.22 -0.62 -1.19 -1.11 -0.67 1.50
2 1 12 10 513 0.07 -0.95 -0.89 -0.85 -0.76 -0.67 1.57
2 1 13 10 517 0.22 -1.70 -1.19 -0.73 -0.83 -0.67 0.57
2 1 14 11 472 0.17 -1.19 -1.13 -0.33 -0.05 -0.67 0.97
2 1 15 11 507 0.19 -1.00 -0.93 -1.17 -0.63 -0.67 0.97
2 1 16 7 531 0.07 -1.58 -1.27 -1.18 -0.96 -0.67 1.57
2 1 17 26 499 0.09 -1.31 -0.89 -0.92 -0.76 -0.67 0.36
2 1 18 23 515 0.07 -0.99 -0.89 -1.19 -1.11 -0.67 0.92
2 1 19 13 520 0.14 -1.56 -0.98 -0.74 -0.76 -0.67 1.34
2 1 20 15 489 0.12 -1.05 -0.89 -0.26 -0.76 -0.67 1.31
2 1 21 16 503 0.18 -1.48 -0.79 -0.73 -0.76 -0.67 0.76
2 1 22 15 518 0.26 -1.37 -1.07 -1.09 -1.07 0.43 0.66
2 1 23 23 500 0.07 -1.03 -0.89 -0.82 -0.76 -0.67 1.06
2 2 1 4 108 0.12 2.31 0.68 -0.12 -0.05 2.64 1.31
2 2 2 3 118 0.11 2.53 0.68 -0.47 -0.76 2.64 0.36
2 2 3 7 99 0.21 1.06 0.80 0.29 1.37 2.64 0.35
2 2 4 2 89 0.06 1.09 2.30 0.25 -0.05 2.64 1.57
2 2 5 7 59 0.13 3.21 0.68 -0.27 1.37 2.64 0.53
2 2 6 7 88 0.11 1.85 0.68 0.16 1.37 2.64 0.85
2 2 7 2 145 0.11 1.14 0.68 0.10 -0.05 2.64 1.36
2 2 8 3 38 0.11 3.32 0.68 -0.27 1.37 2.64 -0.78
2 2 9 6 100 0.17 2.86 0.68 0.10 -0.05 2.64 0.48
2 2 10 2 24 0.11 3.50 2.30 1.23 -0.05 2.64 -0.23
2 2 11 2 12 0.38 4.64 0.68 1.36 0.66 2.64 0.37
2 2 12 6 56 0.25 1.52 2.30 0.32 1.37 2.64 0.77
2 2 13 4 50 0.34 1.64 0.49 1.90 1.37 2.64 0.14
2 2 14 13 76 0.22 2.42 0.68 0.03 1.37 2.64 0.53
2 2 15 4 103 0.12 1.08 2.30 0.14 -0.05 2.64 0.52
2 2 16 11 167 0.13 1.23 0.68 -0.20 -0.05 2.64 0.59
2 2 17 1 107 0 1.99 -0.08 0.49 -0.05 2.64 -0.62
2 2 18 4 84 0.15 1.52 0.68 0.34 1.37 2.64 1.36
2 2 19 5 169 0.17 1.81 -0.08 -0.29 -0.19 2.64 0.99
2 2 20 4 70 0.11 2.69 -0.89 -0.27 1.37 2.64 0.69
2 2 21 4 122 0.11 1.99 0.68 0.22 -0.05 2.64 0.57
2 2 22 3 60 0.12 2.74 -0.89 -0.27 1.37 2.64 -0.78
2 2 23 1 195 0 1.63 0.68 -0.95 -0.76 2.64 0.69
2 3 1 5 39 0.04 1.54 -0.08 2.29 2.79 0.43 -0.81
2 3 2 3 9 0.05 1.71 -0.89 2.29 2.79 2.64 -0.55
2 3 3 10 51 0.07 1.10 -0.08 2.29 2.79 0.43 -0.56
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2 14 13 0 NA NA NA NA NA NA NA NA
2 14 14 0 NA NA NA NA NA NA NA NA
2 14 15 0 NA NA NA NA NA NA NA NA
2 14 16 0 NA NA NA NA NA NA NA NA
2 14 17 0 NA NA NA NA NA NA NA NA
2 14 18 0 NA NA NA NA NA NA NA NA
2 14 19 0 NA NA NA NA NA NA NA NA
2 14 20 0 NA NA NA NA NA NA NA NA
2 14 21 0 NA NA NA NA NA NA NA NA
2 14 22 0 NA NA NA NA NA NA NA NA
2 14 23 0 NA NA NA NA NA NA NA NA
2 15 1 13 341 0.17 0.19 -0.92 -0.52 -0.05 0.43 -0.93
2 15 2 16 361 0.16 0.24 0.68 -0.91 -0.45 -0.67 -1.62
2 15 3 5 456 0.11 -0.29 -0.89 -0.70 -0.76 -0.67 -1.12
2 15 4 26 461 0.2 -0.35 -0.08 -1.04 -0.81 -0.67 -1.59
2 15 5 16 444 0.3 -0.40 0.68 -1.15 -0.94 -0.61 -1.48
2 15 6 15 486 0.08 -0.50 -0.89 -0.92 -0.76 -0.67 -1.66
2 15 7 5 203 0.2 0.69 -1.12 -0.43 1.37 -0.23 -1.58
2 15 8 9 426 0.33 -0.40 -0.75 -0.77 -0.68 0.43 -1.23
2 15 9 14 369 0.19 0.28 -0.92 -0.38 -0.05 -0.67 -0.90
2 15 10 11 464 0.07 -0.06 -0.89 -0.82 -0.76 -0.67 -1.66
2 15 11 5 460 0.11 0.54 -0.89 -0.88 -0.76 -0.67 -1.70
2 15 12 7 410 0.17 0.51 -0.08 -0.92 -0.76 -0.67 -1.57
2 15 13 17 352 0.15 0.35 -0.89 -0.72 -0.13 0.43 -1.66
2 15 14 11 420 0.08 0.02 -1.03 -0.67 -0.05 -0.67 -1.65
2 15 15 13 396 0.13 0.43 -0.89 -0.71 -0.05 -0.67 -1.63
2 15 16 4 379 0.23 0.75 -0.69 -0.65 -0.76 0.15 -0.67
2 15 17 7 322 0.21 0.47 0.24 -0.72 -0.66 0.43 -1.52
2 15 18 25 343 0.22 0.36 -0.08 -0.63 -0.05 -0.67 -1.53
2 15 19 15 496 0.09 -0.38 -1.04 -1.11 -0.76 -0.67 -1.63
2 15 20 14 380 0.3 0.03 -0.08 -0.55 -0.46 -0.67 -0.82
2 15 21 14 293 0.21 0.50 -0.08 -0.67 -0.05 0.43 -1.14
2 15 22 10 353 0.17 0.20 0.68 -0.81 -0.48 -0.67 -0.92
2 15 23 11 432 0.09 -0.35 -0.89 -0.80 -0.05 -0.67 -1.61
2 16 1 8 58 0.25 0.04 2.30 1.70 -0.05 2.64 -1.20
2 16 2 1 33 0 1.25 2.30 1.20 1.37 2.64 -0.94
2 16 3 13 90 0.39 0.07 2.30 0.48 -0.05 2.64 -0.99
2 16 4 5 230 0.11 0.87 -0.89 -0.67 -0.33 2.64 -1.64
2 16 5 11 163 0.15 -0.13 0.68 0.38 -0.05 2.64 -0.78
2 16 6 4 75 0.04 0.84 -0.89 1.20 1.37 2.64 -0.83
2 16 7 17 207 0.2 -0.62 0.68 0.36 -0.76 2.64 -1.01
2 16 8 20 135 0.14 -0.24 0.98 0.51 -0.05 2.64 -1.10
2 16 9 9 132 0.15 1.33 -0.08 -0.65 -0.29 2.64 -1.52
2 16 10 8 74 0.2 0.95 0.39 1.10 1.37 2.64 -0.65
2 16 11 8 284 0.05 -0.76 -0.08 0.51 -0.76 2.64 -1.16
2 16 12 5 83 0.04 0.78 -0.89 1.20 1.37 2.64 -0.43
2 16 13 5 105 0.07 -0.44 0.68 0.49 -0.05 2.64 -2.23
2 16 14 4 136 0.09 1.24 0.68 -0.64 -0.05 2.64 -0.87
2 16 15 2 106 0.03 1.56 0.68 -0.71 -0.76 2.64 -1.71
2 16 16 9 120 0.06 -0.01 0.68 1.23 -0.05 2.64 -0.72
2 16 17 7 67 0.21 0.47 0.41 1.49 1.37 2.64 -0.97
2 16 18 7 213 0.33 0.07 0.57 -0.21 -0.05 2.64 -0.36
2 16 19 5 160 0.15 0.43 0.68 0.62 -0.05 2.64 0.17
2 16 20 6 320 0.13 -0.58 0.68 -0.26 -0.76 2.64 -0.44
2 16 21 3 85 0.06 0.38 0.68 2.29 -0.05 2.64 -0.61
2 16 22 6 345 0.1 -0.82 -0.08 0.21 -0.76 2.64 -0.79
2 16 23 5 110 0.06 1.39 0.68 -0.67 -0.05 2.64 -1.61
2 17 1 6 440 0.09 -0.58 -1.27 0.12 -0.05 -0.67 1.29
2 17 2 10 416 0.13 -0.58 -0.93 -0.18 -0.05 -0.67 0.64
2 17 3 12 417 0.09 -0.24 -0.92 0.15 -0.05 -0.67 1.57
2 17 4 9 349 0.12 0.29 -0.93 0.23 -0.05 -0.67 -0.11
2 17 5 5 309 0.28 0.52 -0.89 0.35 -0.05 0.43 0.52
2 17 6 16 385 0.13 0.31 -0.89 -0.45 -0.05 -0.67 0.41
2 17 7 23 372 0.09 0.31 -0.91 0.03 -0.05 -0.67 0.86
2 17 8 21 403 0.09 -0.15 -1.27 0.09 -0.05 -0.67 0.85
2 17 9 14 366 0.13 -0.37 -0.92 0.45 -0.05 -0.67 -0.31
2 17 10 11 419 0.13 -0.26 -0.89 -0.36 -0.05 -0.67 1.02
2 17 11 13 389 0.11 -0.35 -0.95 0.02 -0.05 -0.67 0.00
2 17 12 17 392 0.06 -0.26 -0.89 0.05 -0.05 -0.67 0.62
2 17 13 4 362 0.17 0.15 -1.08 0.01 -0.05 -0.67 -0.67
2 17 14 9 424 0.17 0.37 -0.89 -0.29 -0.76 -0.67 0.90
2 17 15 12 395 0.09 -0.06 -1.27 -0.09 -0.05 -0.67 0.37
2 17 16 18 399 0.11 -0.17 -0.89 0.09 -0.05 -0.67 1.05
2 17 17 15 373 0.09 -0.59 -0.89 0.45 -0.05 -0.67 -0.58
2 17 18 11 357 0.09 0.20 -0.89 0.47 -0.05 -0.67 0.68
2 17 19 15 398 0.26 0.98 -0.89 -0.41 -0.57 -0.67 0.49
2 17 20 20 367 0.08 0.25 -0.91 0.05 -0.05 -0.67 0.46
2 17 21 3 438 0.08 -0.60 -0.89 0.04 -0.76 -0.67 0.48
2 17 22 6 429 0.08 -0.22 -0.89 -0.07 -0.76 -0.67 0.68
2 17 23 15 382 0.08 -0.35 -0.92 0.45 -0.05 -0.67 0.47
2 18 1 4 430 0.06 -0.12 2.30 -0.78 -0.76 -0.67 0.52
2 18 2 5 199 0.07 0.53 2.30 0.46 -0.05 0.43 0.62
2 18 3 4 276 0.04 -0.21 2.30 0.03 -0.05 -0.67 -0.48
2 18 4 1 225 0 0.74 2.30 -0.19 -0.05 -0.67 -0.79
2 18 5 4 340 0.08 0.42 2.30 -0.56 -0.76 -0.67 -0.64
2 18 6 16 286 0.15 0.56 2.30 0.00 -0.05 -0.67 1.49
2 18 7 5 242 0.04 0.41 2.30 0.45 -0.05 -0.67 0.35
2 18 8 7 187 0.09 0.61 2.30 0.31 -0.05 0.43 1.57
2 18 9 8 200 0.15 0.62 2.30 0.08 -0.05 0.43 0.04
2 18 10 5 226 0.04 0.41 2.30 0.05 -0.05 0.43 0.69
2 18 11 8 463 0.07 0.24 2.30 -0.78 -0.76 -0.67 1.30
2 18 12 9 522 0.08 -0.14 2.30 -1.19 -1.11 -0.67 1.33
2 18 13 2 115 0.08 1.74 2.30 1.23 -0.05 -0.67 -0.03
2 18 14 6 274 0.18 0.31 2.30 -0.06 -0.05 -0.67 0.15
2 18 15 3 229 0.06 0.55 2.30 -0.06 -0.05 0.43 1.14
2 18 16 10 254 0.13 0.67 2.30 0.22 -0.05 -0.67 0.87
2 18 17 4 407 0.13 0.00 2.30 -0.78 -0.76 0.43 0.94
2 18 18 7 246 0.08 0.16 2.30 -0.04 -0.05 0.43 1.57
2 18 19 4 339 0.04 -0.04 2.30 0.05 -0.76 -0.67 0.04
2 18 20 3 217 0.01 0.65 2.30 0.03 -0.05 0.43 0.69
2 18 21 3 476 0.14 -0.52 2.30 -0.45 -0.76 -0.67 1.29
2 18 22 0 NA NA NA NA NA NA NA NA
2 18 23 0 NA NA NA NA NA NA NA NA
2 19 1 6 391 0.18 0.52 -0.89 -0.34 -0.64 0.43 0.98
2 19 2 6 383 0.07 0.68 -0.89 -0.56 -0.76 0.43 0.10
2 19 3 21 470 0.12 -1.02 -0.91 -0.76 -0.76 0.43 0.53
2 19 4 11 348 0.2 0.08 -1.03 -0.10 -0.05 0.43 0.14
2 19 5 10 446 0.15 -1.12 -0.81 -0.30 -0.76 0.43 0.01
2 19 6 14 350 0.08 0.07 -0.89 0.05 -0.05 0.43 0.74
2 19 7 8 388 0.27 -0.53 -0.38 -0.56 -0.40 0.43 -0.28
2 19 8 7 467 0.11 -1.18 -0.89 -0.29 -0.76 0.43 0.88
2 19 9 6 425 0.09 -0.49 -0.89 -0.23 -0.76 0.43 0.58
2 19 10 9 406 0.1 -0.18 -0.08 -0.75 -0.76 0.43 0.63
2 19 11 10 400 0.16 -0.36 -0.89 -0.71 -0.05 0.43 0.33
2 19 12 9 347 0.21 -0.22 -0.89 0.49 -0.13 0.43 0.63
2 19 13 8 497 0.1 -0.83 -0.89 -1.19 -1.11 0.43 0.44
2 19 14 16 378 0.12 -0.25 -1.03 -0.12 -0.05 0.43 0.74
2 19 15 16 418 0.15 -0.73 -0.08 -0.41 -0.76 0.43 0.82
2 19 16 22 448 0.15 -0.21 -0.94 -0.82 -0.76 0.43 0.77
2 19 17 13 485 0.15 -0.74 -0.95 -0.83 -0.76 0.43 1.20
2 19 18 13 447 0.14 -0.52 -0.98 -0.75 -0.76 0.43 0.17
2 19 19 10 412 0.21 -0.48 -0.81 -0.43 -0.05 0.43 1.25
2 19 20 11 355 0.16 -0.61 -0.89 0.20 -0.05 0.43 -0.14
2 19 21 9 390 0.1 -0.66 -0.89 -0.14 -0.05 0.43 0.65
2 19 22 10 401 0.15 -0.52 -0.93 0.13 -0.05 0.43 1.40
2 19 23 11 487 0.12 -0.83 -1.27 -0.86 -0.76 0.43 0.75
2 20 1 5 73 0.02 -0.13 0.68 1.70 1.37 0.43 -2.46
2 20 2 3 69 0.02 -0.02 0.68 2.29 1.37 0.43 -1.98
2 20 3 3 65 0.19 -0.83 1.22 3.07 -0.05 -0.67 -2.46
2 20 4 7 79 0.03 0.82 2.30 1.21 1.37 0.43 -0.92
2 20 5 5 62 0.03 0.13 0.68 2.29 1.37 0.43 -2.32
2 20 6 3 80 0.04 -0.03 0.68 1.70 1.37 0.43 -2.19
2 20 7 3 71 0.03 0.47 2.30 1.70 1.37 0.43 -0.79
2 20 8 2 72 0 0.22 0.68 1.70 1.37 0.43 -2.46
2 20 9 12 35 0.14 0.15 2.30 2.29 1.37 0.43 -2.29
2 20 10 4 44 0.06 -0.03 0.68 3.07 1.37 0.43 -2.25
2 20 11 11 42 0.07 0.15 2.30 1.70 1.37 0.43 -2.39
2 20 12 1 31 0 0.57 2.30 3.30 1.37 0.43 -1.25
2 20 13 5 63 0.08 -0.29 0.68 2.29 1.37 0.43 -2.35
2 20 14 6 66 0.05 0.39 2.30 1.70 1.37 0.43 -1.23
2 20 15 1 68 0 0.31 0.68 2.29 1.37 0.43 -1.98
2 20 16 3 78 0.05 1.06 2.30 1.23 1.37 0.43 -0.62
2 20 17 4 27 0.05 0.19 2.30 3.07 1.37 0.43 -2.25
2 20 18 0 NA NA NA NA NA NA NA NA
2 20 19 0 NA NA NA NA NA NA NA NA
2 20 20 0 NA NA NA NA NA NA NA NA
2 20 21 0 NA NA NA NA NA NA NA NA
2 20 22 0 NA NA NA NA NA NA NA NA
2 20 23 0 NA NA NA NA NA NA NA NA
2 21 1 4 504 0.17 0.72 -0.89 -0.66 -0.76 2.64 1.36
2 21 2 5 245 0.17 0.97 0.68 -0.65 -0.62 2.64 1.03
2 21 3 6 313 0.16 -0.20 0.68 -0.40 -0.52 2.64 0.41
2 21 4 10 338 0.18 0.27 -0.89 -0.34 -0.05 2.64 0.32
2 21 5 4 375 0.13 0.16 -0.49 0.04 -0.76 2.64 0.66
2 21 6 6 298 0.15 1.35 -0.08 -0.45 -0.76 2.64 0.94
2 21 7 10 526 0.21 -0.31 -1.08 -0.88 -0.76 2.64 1.28
2 21 8 11 490 0.15 -0.83 -0.89 -0.25 -0.76 2.64 -0.33
2 21 9 6 393 0.19 1.39 -0.89 -0.56 -0.76 2.64 -0.04
2 21 10 8 359 0.11 0.49 -0.89 -0.39 -0.05 2.64 0.95
2 21 11 6 295 0.1 0.19 -0.08 -0.29 -0.05 2.64 0.75
2 21 12 7 376 0.09 -0.05 -0.89 -0.19 -0.05 2.64 0.77
2 21 13 18 205 0.2 0.58 0.68 -0.13 -0.13 2.64 0.65
2 21 14 6 186 0.18 0.55 0.55 0.01 -0.05 2.64 1.50
2 21 15 6 328 0.14 0.27 -0.89 -0.73 -0.05 2.64 -0.76
2 21 16 2 215 0.1 0.75 -0.08 -0.46 -0.05 2.64 -0.35
2 21 17 5 408 0.12 -0.53 -0.08 -0.29 -0.76 2.64 0.19

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 23 0 0 n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 508 434 0.85 n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 85% of the cells have hit the quantization threshold error in level 2

Let’s plot the Voronoi tessellation for layer 2 (map C)

muHVT::plotHVT(map_C,
        line.width = c(0.4,0.2), 
        color.vec = c("#141B41","#0582CA"),
        centroid.size = 0.1,
        maxDepth = 2) 
Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Heat Maps

Now let’s plot all the features for each cell at level two as a heatmap for better visualization.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "n",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 15: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell

Figure 15: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "price",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 16: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset

Figure 16: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "speed",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 17: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset

Figure 17: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "hd",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 18: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset

Figure 18: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ram",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 19: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset

Figure 19: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "screen",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 20: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset

Figure 20: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ads",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 21: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

Figure 21: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

We now have the set of maps (map A, map B & map C) which will be used to predict which map and cell each test record is assigned to, but before that lets view our test dataset

6 Prediction on Test Data

Now once we have built the model, let us try to predict using our test dataset which cell and which layer each point belongs to.

Raw Testing Dataset

The testing dataset includes the following columns:

Let’s have a look at our randomly selected test dataset containing 1253 datapoints.

Table(head(testComputers_data))
Row.No price speed hd ram screen ads
3 1595 25 170 4 15 94
4 1849 25 170 8 14 94
7 1720 25 170 4 14 94
10 2575 50 210 4 15 94
11 2195 33 170 8 15 94
14 2295 25 245 8 14 94

The predictLayerHVT function is used to score the test data using the predictive set of maps. This function takes an input - a test data and a set of maps (map A, map B, map C).

Now, Let us understand the predictLayerHVT function.

predictLayerHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = T, 
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1, 
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of predictLayerHVT function has been explained below:

The function predicts based on the HVT maps - map A, map B and map C, constructed using HVT function. For each test record, the function will assign that record to Layer1 or Layer2. Layer1 contains the cell ids from map A and Layer 2 contains cell ids from map B (novelty map) and map C (map without novelty).

Prediction Algorithm

The prediction algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the prediction method for a single point in the test dataset:

  1. Calculate the distance between the point and the centroid of all the cells in the first level.
  2. Find the cell whose centroid has minimum distance to the point.
  3. Check if the cell drills down further to form more cells.
  4. If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further.

Note : The prediction algorithm will not work if some of the variables used to perform quantization are missing. In the test dataset, we should not remove any features

Let’s see which cell and layer each point belongs to and check the Mean Absolute Difference.


validation_data <- testComputers
new_predict <- predictLayerHVT(
    data=validation_data,
    map_A,
    map_B,
    map_C,
    normalize = T
  )
predictions <- new_predict %>%
  dplyr::select(Layer1.Cell.ID, Layer2.Cell.ID) %>%
  mutate(Cell.ID = as.numeric(gsub("[A]", "", Layer1.Cell.ID)),
         Row.No = row.names(testComputers_data))

merged_df <- merge(predictions, map_A[[3]]$summary, by = "Cell.ID", all.x = TRUE) %>%
  arrange(as.numeric(Row.No)) %>%
  select(Row.No, price, speed, hd, ram, screen, ads, Layer1.Cell.ID, Layer2.Cell.ID) %>%
  rename(price_act = price, speed_act = speed, hd_act = hd,
         ram_act = ram, screen_act = screen, ads_act = ads)

colnames(merged_df) <- c("Row.No", "price_act", "speed_act", "hd_act", "ram_act", "screen_act", "ads_act",
                         "Layer1.Cell.ID", "Layer2.Cell.ID")

combined <- merged_df %>%
  mutate(Cell.ID = gsub("[C]", "", Layer2.Cell.ID)) %>%
  merge(map_C[[3]]$summary, by = "Cell.ID", all.x = TRUE) %>%
  arrange(as.numeric(Row.No)) %>%
  select(Row.No, price_act, speed_act, hd_act, ram_act, screen_act, ads_act, Layer1.Cell.ID, Layer2.Cell.ID,
         Segment.Level, Segment.Parent, Segment.Child, n, Quant.Error, price, speed, hd, ram, screen, ads)


merged_df <- combined %>%
  mutate(Cell.ID = gsub("[B]", "", Layer2.Cell.ID)) %>%
  merge(map_B[[3]]$summary, by = "Cell.ID", all.x = TRUE) %>%
  arrange(as.numeric(Row.No)) %>%
  select(Row.No, price_act, speed_act, hd_act, ram_act, screen_act, ads_act, Layer1.Cell.ID, Layer2.Cell.ID,
         price.x, speed.x, hd.x, ram.x, screen.x, ads.x, price.y, speed.y, hd.y, ram.y, screen.y, ads.y) %>%
  mutate(price_pred = coalesce(price.x, price.y),
         speed_pred = coalesce(speed.x, speed.y),
         hd_pred = coalesce(hd.x, hd.y),
         ram_pred = coalesce(ram.x, ram.y),
         screen_pred = coalesce(screen.x, screen.y),
         ads_pred = coalesce(ads.x, ads.y)) %>%
  select(Row.No, price_act, speed_act, hd_act, ram_act, screen_act, ads_act, Layer1.Cell.ID, Layer2.Cell.ID,
         price_pred, speed_pred, hd_pred, ram_pred, screen_pred, ads_pred)
sorted_df <- merged_df
sorted_df$diff <- rowMeans(abs(sorted_df[, c("price_act","speed_act","hd_act","ram_act","screen_act","ads_act")] - sorted_df[, c("price_pred","speed_pred","hd_pred","ram_pred","screen_pred","ads_pred")]))
rownames(sorted_df) <- NULL
options(scipen = 999)
sorted_df %>% head(1000) %>%as.data.frame() %>%Table(scroll = T)
Row.No price_act speed_act hd_act ram_act screen_act ads_act Layer1.Cell.ID Layer2.Cell.ID price_pred speed_pred hd_pred ram_pred screen_pred ads_pred diff
1 -0.9674386 -1.0173151 -1.0699653 -0.9362186 0.4307274 -1.4008948 A24 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4419065
2 -0.3449415 -0.8904536 -0.7968915 -0.0501185 -0.6741149 -1.6554312 A143 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2938894
3 -0.6009525 -1.2710382 -1.0771071 -0.7589986 -0.6741149 -1.6630494 A29 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4267005
4 0.4887879 0.1720118 -0.7343040 -0.7589986 0.4307274 -1.5151673 A237 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3912334
5 0.3675560 -0.8904536 -0.7187220 -0.0501185 0.4307274 -1.6549831 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3823913
6 0.0090558 -1.0075566 -0.6679808 -0.0501185 -0.6741149 -1.5682532 A162 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2562455
7 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
8 0.0168493 -0.8904536 -0.6745732 -0.7589986 0.4307274 -1.6944183 A165 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3556972
9 1.0620418 0.1720118 -0.8985638 -0.6408519 -0.6741149 -1.6630494 A214 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4051110
10 1.3306764 -0.0817113 -0.6529316 -0.2864119 2.6404120 -1.5241299 A379 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.6659147
11 3.0346588 -0.0817113 0.1512706 -0.0501185 -0.6741149 -1.6507259 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5705299
12 0.9975291 -1.0173151 -0.2850242 1.3676416 -0.6741149 -1.6059131 A316 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.6435088
13 0.8693078 -0.8904536 -0.6467482 -0.3539243 2.6404120 -1.4466679 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.5724049
14 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
15 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
16 0.4268397 -0.8904536 -0.7057371 -0.0501185 -0.6741149 -1.6271992 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2915539
17 0.4268397 -0.8904536 -0.7057371 -0.0501185 -0.6741149 -1.6271992 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2915539
18 -0.6009525 -1.2710382 -1.0771071 -0.7589986 -0.6741149 -1.6630494 A29 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4267005
19 1.5526376 0.6794579 -0.4507596 -0.3078931 0.4307274 -1.5917564 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3444185
20 -0.9674386 -1.0173151 -1.0699653 -0.9362186 0.4307274 -1.4008948 A24 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4419065
21 -0.8159512 -0.8904536 -1.0411033 -0.7912204 -0.6741149 -1.6455317 A43 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3238867
22 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5427937
23 -1.2826994 -1.2710382 -1.0498386 -0.8298866 -0.6741149 -1.6424355 A10 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3501730
24 0.4268397 -0.8904536 -0.7057371 -0.0501185 -0.6741149 -1.6271992 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2915539
25 -0.3449415 -0.8904536 -0.7968915 -0.0501185 -0.6741149 -1.6554312 A143 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2938894
26 0.3675560 -0.8904536 -0.7187220 -0.0501185 0.4307274 -1.6549831 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3823913
27 0.9975291 -1.0173151 -0.2850242 1.3676416 -0.6741149 -1.6059131 A316 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.5624875
28 -0.8159512 -0.8904536 -1.0411033 -0.7912204 -0.6741149 -1.6455317 A43 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3238867
29 1.5526376 0.6794579 -0.4507596 -0.3078931 0.4307274 -1.5917564 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3444185
30 1.3665387 0.6794579 -0.6639491 -0.1790058 2.6404120 -1.3607671 A386 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.4899168
31 0.0277818 0.6794579 -1.0848169 -0.7811511 -0.6741149 -1.6465247 A166 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3810747
32 1.5526376 0.6794579 -0.4507596 -0.3078931 0.4307274 -1.5917564 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3444185
33 -1.2826994 -1.2710382 -1.0498386 -0.8298866 -0.6741149 -1.6424355 A10 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3501730
34 0.5156322 0.6794579 -0.6943289 -0.0501185 -0.6741149 -1.6079937 A250 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4020764
35 0.2098101 -0.8904536 -0.8031243 -0.7589986 -0.6741149 -1.6966589 A121 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2600096
36 1.0620418 0.1720118 -0.8985638 -0.6408519 -0.6741149 -1.6630494 A214 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4051110
37 0.3675560 -0.8904536 -0.7187220 -0.0501185 0.4307274 -1.6549831 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3823913
38 1.5526376 0.6794579 -0.4507596 -0.3078931 0.4307274 -1.5917564 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3444185
39 1.0620418 0.1720118 -0.8985638 -0.6408519 -0.6741149 -1.6630494 A214 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4051110
40 0.5156322 0.6794579 -0.6943289 -0.0501185 -0.6741149 -1.6079937 A250 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4572695
41 3.0346588 -0.0817113 0.1512706 -0.0501185 -0.6741149 -1.6507259 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5705299
42 0.8575062 0.6794579 -0.6465256 -0.1210066 0.4307274 -1.4053761 A292 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4306902
43 -0.2117972 -0.0817113 -1.0117870 -0.7744091 -0.6741149 -1.6435656 A126 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2803545
44 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
45 0.5156322 0.6794579 -0.6943289 -0.0501185 -0.6741149 -1.6079937 A250 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4020764
46 0.9975291 -1.0173151 -0.2850242 1.3676416 -0.6741149 -1.6059131 A316 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.5624875
47 1.5474644 0.6794579 -0.3757335 1.3676416 0.4307274 -1.0222272 A370 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4172149
48 0.8693078 -0.8904536 -0.6467482 -0.3539243 2.6404120 -1.4466679 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.5724049
49 -1.2106241 -0.8904536 -1.2808087 -1.1134387 -0.6741149 -1.6372821 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3596245
50 3.0346588 -0.0817113 0.1512706 -0.0501185 -0.6741149 -1.6507259 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5705299
51 3.0339927 0.6794579 0.2297796 -0.0501185 -0.4191513 -1.6840769 A382 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4957143
52 0.8693078 -0.8904536 -0.6467482 -0.3539243 2.6404120 -1.4466679 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.5724049
53 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
54 0.5156322 0.6794579 -0.6943289 -0.0501185 -0.6741149 -1.6079937 A250 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4572695
55 1.6169873 0.3532425 -0.2850242 1.3676416 -0.6741149 -1.6521663 A366 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3337007
56 -0.3540405 -0.8904536 -0.9781217 -0.7589986 -0.6741149 -1.6344382 A85 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3241995
57 0.0090558 -1.0075566 -0.6679808 -0.0501185 -0.6741149 -1.5682532 A162 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2562455
58 -0.9674386 -1.0173151 -1.0699653 -0.9362186 0.4307274 -1.4008948 A24 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4419065
59 0.5156322 0.6794579 -0.6943289 -0.0501185 -0.6741149 -1.6079937 A250 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4020764
60 1.5474644 0.6794579 -0.3757335 1.3676416 0.4307274 -1.0222272 A370 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4172149
61 0.5156322 0.6794579 -0.6943289 -0.0501185 -0.6741149 -1.6079937 A250 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4572695
62 -0.6009525 -1.2710382 -1.0771071 -0.7589986 -0.6741149 -1.6630494 A29 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4267005
63 -0.3540405 -0.8904536 -0.9781217 -0.7589986 -0.6741149 -1.6344382 A85 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3241995
64 -0.2117972 -0.0817113 -1.0117870 -0.7744091 -0.6741149 -1.6435656 A126 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2803545
65 0.8575062 0.6794579 -0.6465256 -0.1210066 0.4307274 -1.4053761 A292 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4306902
66 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
67 3.0346588 -0.0817113 0.1512706 -0.0501185 -0.6741149 -1.6507259 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5705299
68 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
69 1.6169873 0.3532425 -0.2850242 1.3676416 -0.6741149 -1.6521663 A366 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3337007
70 0.0277818 0.6794579 -1.0848169 -0.7811511 -0.6741149 -1.6465247 A166 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3810747
71 -1.2106241 -0.8904536 -1.2808087 -1.1134387 -0.6741149 -1.6372821 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3596245
72 1.6169873 0.3532425 -0.2850242 1.3676416 -0.6741149 -1.6521663 A366 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3337007
73 0.0168493 -0.8904536 -0.6745732 -0.7589986 0.4307274 -1.6944183 A165 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3556972
74 -0.2117972 -0.0817113 -1.0117870 -0.7744091 -0.6741149 -1.6435656 A126 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2803545
75 0.6126177 -0.0817113 -0.6667822 -0.0501185 0.4307274 -1.3986542 A261 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3651749
76 0.3675560 -0.8904536 -0.7187220 -0.0501185 0.4307274 -1.6549831 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3823913
77 0.0277818 0.6794579 -1.0848169 -0.7811511 -0.6741149 -1.6465247 A166 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3810747
78 0.4887879 0.1720118 -0.7343040 -0.7589986 0.4307274 -1.5151673 A237 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3912334
79 0.0090558 -1.0075566 -0.6679808 -0.0501185 -0.6741149 -1.5682532 A162 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2562455
80 -1.2106241 -0.8904536 -1.2808087 -1.1134387 -0.6741149 -1.6372821 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3596245
81 -1.2106241 -0.8904536 -1.2808087 -1.1134387 -0.6741149 -1.6372821 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3596245
82 0.2098101 -0.8904536 -0.8031243 -0.7589986 -0.6741149 -1.6966589 A121 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2600096
83 -0.4639412 -0.9750280 -0.8736759 -0.7589986 -0.6741149 -1.1163339 A104 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2751763
84 0.8225694 -1.0288480 0.1512706 1.3676416 -0.6741149 -0.8450132 A301 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3463495
85 0.0168493 -0.8904536 -0.6745732 -0.7589986 0.4307274 -1.6944183 A165 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3556972
86 -0.5366611 -0.8904536 -0.7555993 -0.0501185 -0.6741149 -0.7331850 A142 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4015662
87 0.6126177 -0.0817113 -0.6667822 -0.0501185 0.4307274 -1.3986542 A261 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3651749
88 0.3343904 -0.0817113 -0.6924924 -0.0501185 -0.6741149 -1.6366099 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2500833
89 0.3305203 -0.9782808 -0.2940138 -0.0501185 -0.6741149 -0.8484916 A201 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4182050
90 0.8693078 -0.8904536 -0.6467482 -0.3539243 2.6404120 -1.4466679 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.5724049
91 0.8334330 -0.4860824 -0.6550957 1.3676416 0.4307274 -1.5868677 A345 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.7396201
92 0.4268397 -0.8904536 -0.7057371 -0.0501185 -0.6741149 -1.6271992 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2915539
93 0.8575062 0.6794579 -0.6465256 -0.1210066 0.4307274 -1.4053761 A292 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.5715353
94 0.3234717 -0.9596508 -0.4443852 -0.0501185 0.4307274 -0.8902333 A226 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4801580
95 2.4596730 0.6794579 0.2129491 -0.0501185 -0.6741149 -0.8676232 A337 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4491331
96 -0.5366611 -0.8904536 -0.7555993 -0.0501185 -0.6741149 -0.7331850 A142 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4015662
97 0.4887879 0.1720118 -0.7343040 -0.7589986 0.4307274 -1.5151673 A237 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3912334
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513 1.2131488 -0.8904536 0.4560926 1.3676416 -0.6741149 0.8607481 A299 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1683354
514 0.6345812 -1.0288480 -0.8480995 -0.7589986 2.6404120 0.9613476 A4 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3593146
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521 1.1483887 0.6794579 -0.2058160 -0.1176309 -0.6741149 0.7920486 A251 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2205290
522 0.8732664 -1.2710382 0.4606837 1.3676416 -0.6741149 0.8637490 A290 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1763845
523 -1.2212752 -1.1949213 -0.9184308 -0.7589986 0.4307274 0.9056169 A38 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2263641
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526 1.1483887 0.6794579 -0.2058160 -0.1176309 -0.6741149 0.7920486 A251 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2205290
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529 0.7009785 0.6794579 0.1645152 -0.0501185 -0.6741149 0.6087773 A253 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1620490
530 0.4535729 0.6794579 0.0446083 -0.1176309 2.6404120 0.8554266 A336 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.4808829
531 0.9382121 -0.0817113 -0.3989673 -0.1387286 0.4307274 0.4305458 A252 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3033670
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533 1.3839800 0.6794579 -0.2133905 1.3676416 0.4307274 0.4177555 A355 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2780810
534 -1.0612491 -1.0807459 -0.3831906 -0.0501185 -0.6741149 0.6044753 A115 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.3153555
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539 0.9595457 -0.8904536 0.4558270 1.3676416 0.4307274 0.7242475 A323 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2184596
540 -1.2106241 -1.2710382 -1.1926732 -1.1134387 -0.6741149 0.9685788 A18 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1741589
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543 1.3475142 0.6794579 -0.4096799 -0.2273386 0.4307274 0.9013597 A280 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3151982
544 0.4436627 0.7508175 0.0755249 -0.0501185 -0.6741149 1.0526027 A239 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1731988
545 -0.7573899 -0.0817113 -0.1424493 -0.7235546 -0.6741149 0.7422744 A132 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2910648
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547 0.4609046 -0.8904536 -0.2850242 -0.7589986 -0.6741149 0.7523573 A144 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2951347
548 0.7009785 0.6794579 0.1645152 -0.0501185 -0.6741149 0.6087773 A253 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1620490
549 0.6386747 -0.0817113 -0.0496423 -0.0501185 -0.6741149 0.6753609 A221 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1591266
550 1.5436391 0.6794579 0.4653069 1.3676416 -0.6741149 0.8743858 A353 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2459232
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552 1.3839800 0.6794579 -0.2133905 1.3676416 0.4307274 0.4177555 A355 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2780810
553 -0.7842712 -0.8904536 -0.7624999 -0.7589986 -0.6741149 0.9636174 A72 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1732465
554 -0.6811459 -0.0817113 -0.4568358 -0.7589986 0.4307274 0.7891509 A149 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2421214
555 -0.4780113 0.6794579 -0.0952910 -0.1335162 0.4307274 0.5989396 A233 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1657750
556 -0.4977567 0.6794579 -1.1845281 -1.0812168 -0.6741149 0.4834809 A112 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.1920232
557 0.6345812 -1.0288480 -0.8480995 -0.7589986 2.6404120 0.9613476 A4 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3593146
558 -1.2294343 -0.8904536 -0.7529158 -0.7589986 -0.6741149 0.6522755 A63 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1176381
559 0.4535729 0.6794579 0.0446083 -0.1176309 2.6404120 0.8554266 A336 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3884291
560 -0.2720868 -1.2710382 -0.8070198 -0.7589986 -0.6741149 0.8091201 A84 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1409427
561 -0.6183813 0.6794579 -0.1470126 -0.7589986 -0.6741149 0.8387819 A154 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2450028
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563 -0.7238201 -0.8904536 -0.0668768 -0.0501185 0.4307274 0.7187477 A176 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2031131
564 0.6345812 -1.0288480 -0.8480995 -0.7589986 2.6404120 0.9613476 A4 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3593146
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566 0.1416116 -0.0817113 -0.7464899 -0.4863524 0.4307274 0.6654895 A191 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.2775415
567 0.0881756 0.6794579 -0.3610888 -0.0501185 0.4307274 0.6504673 A247 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1174805
568 0.0882030 -0.0817113 0.0215508 -0.0501185 -0.6741149 0.8599079 A204 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.2165641
569 -0.4622643 -0.0817113 -0.8218598 -0.6239738 -0.6741149 0.6960213 A120 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.1499995
570 -0.7573899 -0.0817113 -0.1424493 -0.7235546 -0.6741149 0.7422744 A132 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2910648
571 1.5436391 0.6794579 0.4653069 1.3676416 -0.6741149 0.8743858 A353 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2459232
572 0.2863498 -0.8904536 0.4473277 1.3676416 -0.6741149 0.6582506 A264 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1680307
573 1.5436391 0.6794579 0.4653069 1.3676416 -0.6741149 0.8743858 A353 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.2459232
574 0.4233612 -1.1441767 0.4525217 1.3676416 0.4307274 1.0167525 A307 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.3632843
575 0.7009785 0.6794579 0.1645152 -0.0501185 -0.6741149 0.6087773 A253 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1620490
576 -0.6183813 0.6794579 -0.1470126 -0.7589986 -0.6741149 0.8387819 A154 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2450028
577 -0.9304917 -0.8904536 -1.1420319 -0.7589986 -0.6741149 0.6851382 A59 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1498168
578 -0.6319890 -0.8904536 -0.7914378 -0.7589986 -0.6741149 0.5795082 A90 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.0490318
579 -1.3233935 -1.0288480 -1.1926732 -1.1134387 0.4307274 0.8806847 A15 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2792940
580 -0.6679957 -0.8904536 -1.0978830 -0.7589986 -0.6741149 0.3911666 A75 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1080114
581 0.2863498 -0.8904536 0.4473277 1.3676416 -0.6741149 0.6582506 A264 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.1680307
582 0.2385718 -0.0817113 -0.0911773 -0.0501185 0.4307274 0.6857784 A238 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1251137
583 0.9864884 -0.0817113 0.4473277 1.3676416 -0.6741149 0.7153868 A314 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.2171907
584 -0.6081447 -0.8904536 -0.2071145 -0.7589986 0.4307274 0.6246410 A141 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.1189319
585 -0.8774768 -0.8904536 -0.7775872 -0.7589986 -0.6741149 0.6612381 A77 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1058765
586 -1.4998883 -0.8904536 -0.7531912 -0.7589986 -0.6741149 0.6179191 A54 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1546192
587 0.0881756 0.6794579 -0.3610888 -0.0501185 0.4307274 0.6504673 A247 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1174805
588 -1.1819614 -0.9285121 -0.1884161 -0.7589986 -0.6741149 0.6582506 A89 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.2122946
589 -0.1592834 0.6794579 0.0325880 -0.0501185 -0.6741149 0.7631124 A216 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.1944648
590 -0.8841020 -0.0817113 -1.1926732 -1.1134387 -0.6741149 0.9069613 A50 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.3394324
591 -0.1760634 -0.9750280 -0.8459747 -0.7589986 0.4307274 0.6806569 A122 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.2012773
592 0.0882030 -0.0817113 0.0215508 -0.0501185 -0.6741149 0.8599079 A204 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2405346
593 -1.0908643 -1.2710382 -0.8132526 -0.7589986 -0.6741149 0.5291899 A55 C471 -0.5129045 -0.9728948 -0.8664462 -0.7547916 -0.6741149 0.5660586 0.1617288
594 -0.3664620 -0.8904536 -0.0384228 -0.0501185 -0.6741149 0.6728634 A157 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.1189674
595 1.5188746 -0.5310126 -0.5473205 -0.7589986 2.6404120 0.3565114 A311 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3522196
596 -1.6648110 -1.2710382 -0.8420792 -0.7589986 -0.6741149 0.8431031 A22 C512 -1.2707584 -1.0397430 -0.9098567 -0.8315183 -0.5949957 0.8589108 0.1434286
597 0.8713613 2.2969425 0.2132088 -0.0501185 -0.6741149 1.3116203 A317 C301 0.3539798 2.2969425 -0.1275540 -0.2934206 -0.2808659 0.8171839 0.3315220
598 -0.5081283 1.0540959 -1.1926732 -1.1134387 -0.6741149 1.3573293 A30 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3483079
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788 0.6604177 0.6794579 0.5969145 1.3676416 -0.6741149 0.4162618 A327 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.3585708
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791 0.6604177 0.6794579 0.5969145 1.3676416 -0.6741149 0.4162618 A327 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.3585708
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795 1.6918707 0.6794579 2.2859986 2.7854017 -0.6741149 0.3837725 A417 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.4096297
796 0.3091667 0.6250887 0.1409753 -0.0501185 0.4307274 0.0696032 A273 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.1881152
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802 4.0729238 1.4882002 1.2936228 0.3043215 2.6404120 0.0700833 A434 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.8026677
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905 -0.6073378 0.6794579 -0.0243806 -0.7589986 -0.6741149 0.1718286 A174 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3159403
906 2.0420963 2.2969425 0.9113238 -0.0501185 -0.3058341 -0.3395798 A393 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.6638782
907 1.3313500 -0.8904536 2.2859986 2.7854017 0.4307274 -0.3020224 A416 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.2733065
908 2.0420963 2.2969425 0.9113238 -0.0501185 -0.3058341 -0.3395798 A393 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.6638782
909 0.8082245 -0.8904536 1.1952617 1.3676416 2.6404120 -0.6084562 A401 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.8542774
910 -0.5804512 0.1524946 -0.8864278 -0.8680571 0.4307274 -0.5785811 A164 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.5196174
911 -1.4853571 -0.9128409 -0.3010645 -0.7589986 -0.6741149 -0.2037357 A81 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2444403
912 -1.3283668 -0.8904536 -0.8125193 -0.7589986 -0.6741149 -0.5145252 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1150137
913 -0.9967363 0.6794579 0.0767693 -0.7589986 -0.6741149 -0.4575867 A182 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3210328
914 1.3313500 -0.8904536 2.2859986 2.7854017 0.4307274 -0.3020224 A416 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.2733065
915 -0.8302786 -0.8904536 -0.2538603 -0.7589986 2.6404120 -0.3268150 A13 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.4810704
916 0.1091055 0.6794579 1.4769355 -0.0501185 2.6404120 -0.6106394 A387 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.2369801
917 -1.2806211 -1.0173151 -0.2850242 -0.0501185 -0.6741149 -0.2985014 A124 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3316817
918 -0.4305281 -0.8904536 -0.7454002 -0.0501185 -0.6741149 0.3331545 A133 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2587646
919 4.0729238 1.4882002 1.2936228 0.3043215 2.6404120 0.0700833 A434 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.8026677
920 -0.8757931 -0.9221690 -0.7765051 -0.7589986 -0.6741149 -0.4127739 A93 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1581995
921 -0.6073378 0.6794579 -0.0243806 -0.7589986 -0.6741149 0.1718286 A174 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3159403
922 -0.7924358 -0.8904536 0.4473277 -0.0501185 -0.6741149 -0.6832508 A186 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.4219462
923 -1.1017205 -0.8904536 -0.8144007 -0.7589986 -0.6741149 -0.8962798 A74 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1082291
924 -0.8114247 -0.0817113 0.3027077 -0.0501185 0.4307274 -0.6491611 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2678296
925 -0.1827985 -0.8904536 2.2859986 -0.0501185 -0.6741149 -0.3814050 A258 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.5764775
926 0.0277917 0.5410635 2.2859986 -0.0501185 -0.6741149 -0.3573691 A319 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.6062010
927 -0.6607113 0.6794579 -0.7840010 -0.7912204 -0.6741149 -0.7325739 A150 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3836774
928 0.2674915 -0.0817113 1.1952617 1.3676416 0.4307274 -0.7488695 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2100719
929 -0.4192393 -0.8904536 0.4473277 -0.0501185 -0.6741149 -0.4112055 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3144059
930 -1.2806211 -1.0173151 -0.2850242 -0.0501185 -0.6741149 -0.2985014 A124 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3316817
931 -1.7516938 -0.9221690 -0.6187378 -0.7589986 -0.6741149 -0.8799467 A41 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1865311
932 0.2307371 0.7046437 1.2113019 1.3676416 0.4307274 -0.7438610 A372 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3187495
933 -0.6162509 0.6794579 0.0341895 -0.0501185 -0.6741149 -0.6510283 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2237006
934 0.6009959 0.6794579 1.1659430 1.3676416 0.4307274 -0.7045285 A376 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.6122255
935 -1.3016347 -0.0817113 0.2551503 -0.7589986 0.4307274 -0.9039216 A206 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.4054436
936 1.4079465 -0.8904536 2.2859986 2.7854017 0.4307274 -0.8217920 A419 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3471686
937 -0.4157831 0.6794579 0.4591836 -0.0501185 -0.6741149 -0.6270957 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2514902
938 -0.4192393 -0.8904536 0.4473277 -0.0501185 -0.6741149 -0.4112055 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3144059
939 -0.6162509 0.6794579 0.0341895 -0.0501185 -0.6741149 -0.6510283 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2237006
940 -0.2770516 0.6794579 0.3247497 -0.0501185 0.4307274 -0.7399069 A276 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2495755
941 0.0277917 0.5410635 2.2859986 -0.0501185 -0.6741149 -0.3573691 A319 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.6062010
942 1.0390496 0.6794579 2.2859986 2.7854017 0.4307274 -0.8442124 A421 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3711582
943 -0.6980481 0.6794579 0.4256243 -0.0501185 0.4307274 -0.7773576 A272 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.1899801
944 -0.5246534 0.6794579 1.2309703 -0.0501185 0.4307274 -0.8317731 A303 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.5630211
945 -0.7924358 -0.8904536 0.4473277 -0.0501185 -0.6741149 -0.6832508 A186 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.4017059
946 -0.6607113 0.6794579 -0.7840010 -0.7912204 -0.6741149 -0.7325739 A150 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.4050144
947 -1.7516938 -0.9221690 -0.6187378 -0.7589986 -0.6741149 -0.8799467 A41 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1865311
948 0.4056575 -0.8904536 0.2642397 -0.0501185 -0.6741149 -0.3410735 A217 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2819138
949 -0.5085304 -0.8904536 0.4473277 -0.0501185 0.4307274 -0.4729510 A227 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.4933212
950 -0.2770516 0.6794579 0.3247497 -0.0501185 0.4307274 -0.7399069 A276 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2495755
951 -0.4192393 -0.8904536 0.4473277 -0.0501185 -0.6741149 -0.4112055 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3144059
952 0.2404356 -0.8904536 1.1952617 1.3676416 0.4307274 -0.8393912 A358 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3644585
953 -1.2334850 -0.9285121 -0.8494806 -0.8298866 0.4307274 -0.4562423 A86 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2671521
954 0.6256068 -0.8904536 1.1952617 1.3676416 0.4307274 -0.7207015 A361 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3947997
955 -0.6162509 0.6794579 0.0341895 -0.0501185 -0.6741149 -0.6510283 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2237006
956 0.1091055 0.6794579 1.4769355 -0.0501185 2.6404120 -0.6106394 A387 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.2369801
957 1.0390496 0.6794579 2.2859986 2.7854017 0.4307274 -0.8442124 A421 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3711582
958 -0.6607113 0.6794579 -0.7840010 -0.7912204 -0.6741149 -0.7325739 A150 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3836774
959 0.2675685 -0.8904536 1.1952617 1.3676416 0.4307274 -0.4486241 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3511759
960 0.2675685 -0.8904536 1.1952617 1.3676416 0.4307274 -0.4486241 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3511759
961 1.0390496 0.6794579 2.2859986 2.7854017 0.4307274 -0.8442124 A421 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3711582
962 -1.3283668 -0.8904536 -0.8125193 -0.7589986 -0.6741149 -0.5145252 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1150137
963 -0.5085304 -0.8904536 0.4473277 -0.0501185 0.4307274 -0.4729510 A227 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.4125301
964 -0.6162509 0.6794579 0.0341895 -0.0501185 -0.6741149 -0.6510283 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2237006
965 0.1091055 0.6794579 1.4769355 -0.0501185 2.6404120 -0.6106394 A387 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.2369801
966 0.6009959 0.6794579 1.1659430 1.3676416 0.4307274 -0.7045285 A376 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3599608
967 -0.4192393 -0.8904536 0.4473277 -0.0501185 -0.6741149 -0.4112055 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3144059
968 -1.3283668 -0.8904536 -0.8125193 -0.7589986 -0.6741149 -0.5145252 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1150137
969 -0.8757931 -0.9221690 -0.7765051 -0.7589986 -0.6741149 -0.4127739 A93 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1581995
970 -0.4157831 0.6794579 0.4591836 -0.0501185 -0.6741149 -0.6270957 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2514902
971 -0.4157831 0.6794579 0.4591836 -0.0501185 -0.6741149 -0.6270957 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2514902
972 -0.2791298 0.2227564 2.2859986 -0.0501185 0.4307274 -0.5463159 A339 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.4623280
973 -0.8114247 -0.0817113 0.3027077 -0.0501185 0.4307274 -0.6491611 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2678296
974 -1.2282677 -0.0817113 -0.5706123 -0.7589986 -0.6741149 -0.3735628 A116 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2266976
975 0.2675685 -0.8904536 1.1952617 1.3676416 0.4307274 -0.4486241 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3511759
976 0.1662244 0.6794579 -0.6268534 -0.0501185 -0.6741149 -0.8491380 A229 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.4482497
977 1.3313500 -0.8904536 2.2859986 2.7854017 0.4307274 -0.3020224 A416 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.2733065
978 -0.3078791 0.6794579 -0.9506660 -0.8919136 -0.6741149 -0.1366154 A146 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.2564913
979 -0.8114247 -0.0817113 0.3027077 -0.0501185 0.4307274 -0.6491611 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2678296
980 0.0277917 0.5410635 2.2859986 -0.0501185 -0.6741149 -0.3573691 A319 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.6062010
981 -1.3283668 -0.8904536 -0.8125193 -0.7589986 -0.6741149 -0.5145252 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1150137
982 0.2674915 -0.0817113 1.1952617 1.3676416 0.4307274 -0.7488695 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2100719
983 -0.8757931 -0.9221690 -0.7765051 -0.7589986 -0.6741149 -0.4127739 A93 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1581995
984 0.9212868 2.2969425 1.2165098 1.3676416 0.4307274 -0.7936822 A403 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.5776074
985 -0.8114247 -0.0817113 0.3027077 -0.0501185 0.4307274 -0.6491611 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2678296
986 -1.3267470 -0.8904536 0.0608952 -0.7589986 -0.6741149 -0.6538665 A111 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1995789
987 -1.3267470 -0.8904536 0.0608952 -0.7589986 -0.6741149 -0.6538665 A111 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1995789
988 0.1133686 0.6794579 0.1407827 -0.0501185 2.6404120 -0.3893334 A344 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.1640285
989 -1.3283668 -0.8904536 -0.8125193 -0.7589986 -0.6741149 -0.5145252 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1150137
990 -0.4192393 -0.8904536 0.4473277 -0.0501185 -0.6741149 -0.4112055 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3144059
991 0.2674915 -0.0817113 1.1952617 1.3676416 0.4307274 -0.7488695 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2100719
992 0.2674915 -0.0817113 1.1952617 1.3676416 0.4307274 -0.7488695 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2100719
993 0.2675685 -0.8904536 1.1952617 1.3676416 0.4307274 -0.4486241 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3511759
994 0.2675685 -0.8904536 1.1952617 1.3676416 0.4307274 -0.4486241 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3511759
995 -0.4157831 0.6794579 0.4591836 -0.0501185 -0.6741149 -0.6270957 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2514902
996 -1.3267470 -0.8904536 0.0608952 -0.7589986 -0.6741149 -0.6538665 A111 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1995789
997 -0.5085304 -0.8904536 0.4473277 -0.0501185 0.4307274 -0.4729510 A227 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.4125301
998 -1.1499216 0.6794579 -0.7889058 -0.7589986 -0.6741149 -0.4031392 A129 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3398255
999 -0.7834052 -0.0817113 0.3833304 -0.7589986 2.6404120 -0.9991806 A295 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.4201735
1000 -1.1499216 0.6794579 -0.7889058 -0.7589986 -0.6741149 -0.4031392 A129 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3398255
hist(sorted_df$diff, breaks = 20, col = "blue", main = "Mean Absolute Difference", xlab = "Difference")
Figure 22: Mean Absolute Difference

Figure 22: Mean Absolute Difference

7 Executive Summary

8 References

  1. Topology Preserving Maps : https://users.ics.aalto.fi/jhollmen/dippa/node9.html#:~:text=The%20property%20of%20topology%20preserving,tool%20of%20high-dimensional%20data

  2. Vector Quantization : https://en.wikipedia.org/wiki/Vector_quantization

  3. K-means : https://en.wikipedia.org/wiki/K-means_clustering

  4. Sammon’s Projection : http://en.wikipedia.org/wiki/Sammon_mapping

  5. Voronoi Tessellations : http://en.wikipedia.org/wiki/Centroidal_Voronoi_tessellation